and so substituting and reducing, we obtain the following expressions for the nine cosines:

These functions can, of course, be expanded in series by the formulæ given in section 10, Part II. of this Report. Jacobi in his memoir enters into a discussion of the ambiguities occasioned by the use of the symbol i= √−1, which I omit here, my object being to give a clear insight into the principle of the method by which the problem of the motion of a rigid body round a fixed point is solved.

Section 3.-In the 50th volume of Crelle's Journal there is a memoir by Lottner on the motion of a rigid solid of revolution round a fixed point which is not its centre of gravity, but which is situated in the axis of revolution. This memoir is very similar in its character to Jacobi's. I shall content myself therefore with giving results.

The equations of motion are given by Poisson in the following form:

* These values of cos 0 and cos are of course derived from equations (3).

where A and C are the moments round the a, and z, axes, z, being the axis of revolution,

γ

the distance of the centre of gravity from the plane of x, y1,

(n) the constant angular velocity round the axis of revolution,

(1) the moment of the quantity of motion of all the points of the body relative to the vertical axis of z,

(h) a quantity introduced by the integration.

Then if a, a,, a, are the three roots of the cubic equation,

(2APy+Ah) (1 — §3) — (CnE —7)2=0;

when-a, is greater than unity, and a,, a, lie between 1 and +1, k the

H(¿(a1+a1)+K) H(i(a,—a2)—K)=D,

→ (u—ia,)=A ̧, ¤(u+ia,)=B',

O(u—ia,—K)=A", 0(u+ia,+K)=B".

Then a, a', a", 6, &c. being the same nine direction cosines as before,

1 H3ia,(B"2+A"2)—H2(ia,+K) (B'2+A'3)

where the axis of (a) revolves about the azis of z with an angular velocity

and the axis of x, round the axis of z, with an angular velocity

n(A-C) d log, Hia, _dlog, H(ia, +K)}.

A

-m

{

da,

da2

There is also, in the 50th volume of Crelle's Journal, an elaborate memoir on the application of the functions to the solution of the problem of ascertaining the motion of the spherical pendulum, by Dumas.

Section 4.-It will be interesting, in writing on elliptic functions in a country so dependent for its greatness, under Providence, upon its manufacturing skill as this, to show that these integrals are capable of a direct application to machinery. A remarkable example of this is given by Canon Moseley in his 'Mechanics.'

The quantity of work done by a pressure P acting through a space S, where P and S are constant, is taken to be equal to PS. Hence if P is

variable, the work done is equal to PdS, or half the vis viva accumulated

while the work is being done.

SP

2

Canon Moseley then shows that in any machine, if U, is the work done at its moving point through the space S, U, the work yielded at the working points, U, and U, are connected together by an equation of the form U,AU,+BS, where A and B are constants dependent for their value upon the construction of the machine, that is to say, upon the dimensions and combination of its parts, their weights, and the coefficients of friction at the various rubbing-surfaces. Upon this principle Canon Moseley works out his theory, and the above equation is applied to the wheel and axle, to pulleys combined in different ways, to toothed wheels, and to all the component parts of machinery, affording in many cases, and especially with regard to toothed wheels, results of great interest and beauty.

In the case of the capstan, the above equation leads to an elliptic function. Let a1 be the length of the lever turning the capstan measured from the axis ;

a, the length of the perpendicular upon the rope supposed to act in a constant direction;

T the tension of the rope;

U, the work done by the pressure applied to the extremity of the lever always perpendicular to its direction;

U, the work actually performed by the capstan;

ρ the radius of the axle, and the limiting angle of resistance. For a full explanation of this latter quantity I must refer the reader to the original treatise.

Then if T be supposed constant,

This integral is, of course, the elliptic function E; and the result is strongly suggestive of the importance of the higher integrals in a calculation of work done in machines, when the point of application of the motive power is variable. It is hardly necessary to observe that the radical in the above integral gives the distance between the points of application of the forces.

Committee for the purpose of promoting the extension, improvement, and harmonic analysis of Tidal Observations. Consisting of Sir WILLIAM THOMSON, LL.D., F.R.S., Prof. J. C. ADAMS, F.R.S., The ASTRONOMER ROYAL, F.R.S., J. F. BATEMAN, F.R.S., Admiral Sir EDWARD BELCHER, K.C.B., T. G. BUNT, Staff-Commander BURDWOOD, R.N., WARREN DE LA RUE, F.R.S., Prof. Fischer, F.R.S., J. P. GASSIOT, F.R.S., Prof. HAUGHTON, F.R.S., J. R. HIND, F.R.S., Prof. KELLAND, F.R.S., Staff-Captain MORIARTY, C.B., J. OLDHAM, C.E., W. PARKES, M. Inst. C.E., Prof. B. PRICE, F.R.S., Rev. C. PRITCHARD, LL.D., F.R.S., Prof. RANKINE, LL.D., F.R.S., Captain RICHARDS, R.N., F.R.S., Dr. ROBINSON, F.R.S., General SABINE, President of the Royal Society, W. SISSONS, Prof. STOKES, D.C.L., F.R.S., T. WEBSTER, M.A., F.R.S., and Prof. FULLER, M.A., and J. F. ISELIN, M.A., Secretaries.

41. THE Committee have to report that the superintendence of the work for the past year has been wholly undertaken by Sir William Thomson. That work has consisted in the reduction of observations and determination of constants by Mr. Roberts and assistants, according to the method which has been fully described in the Report of 1868. For the details of the results obtained, the Committee beg leave to refer to the statements by Sir William Thomson which are appended hereto.

Exeter, August 1869.

W. PARKES.

GEORGE HENRY RICHARDS.
W. J. MACQUORN RANKINE.
J. C. ADAMS.

Report for 1869 by Sir W. THOMSON.

42. From the Meeting of the British Association at Norwich (Aug. 1868) up to the present time the harmonic reduction of observations recorded by self-registering tide-gauges in several different localities, namely Ramsgate, Bombay, Liverpool, and Fort Point, California, has been continued. The

work has been performed by Mr. E. Roberts and assistant calculators in the Nautical Almanac Office, working under his immediate direction, according to the plans described in the Report presented by the Committee of 1867-68 to the Association at Norwich a year ago, with modifications suggested by experience, and extensions to include parts of the investigation not reached in the first year's work. The results obtained up to January 1869 are described in a supplement to that report, which has been printed, and is published in the yearly volume of the Association.

43. The long-period tides, shown in §§ 28, 29 of this supplement, that is to say, the lunar monthly (elliptic), the lunar fortnightly (declinational), the solar annual, and the solar semiannual, were calculated in consequence of the astronomical anticipation of the existence of such tides indicated in the general schedule of § 2 of the first Report. There is a mistake in the argument printed for the lunar monthly, which has been pointed out to me by Mr. Roberts. It ought to be (-) t, instead of a t. The error produces scarcely a sensible influence on the calculations which have been made, and it is easily allowed for.

44. The "luni-solar fortnightly shallow-water (synodic) tide" is a tide the existence of which was suggested by Helmholtz's theory of compound sounds (§§ 24, 25 of first Report). The harmonic analysis consequently applied to discover it has proved it to be very sensible both at Ramsgate and Liverpool; and has shown that in each station it gives highest average level at the times of neap-tides, and lowest average at the times of springtides. Its amount for Ramsgate (§ 28) is a tenth of a foot above and below the mean level. Its amount at Liverpool is rather less, being only sevenhundredths of a foot above and below mean level, as will be seen later.

45. It will be seen that the lunar declinational fortnightly and the solar (declinational or meteorological) semiannual present no agreement with astronomical theory. The solar is of more than twice the amount of the lunar. The lunar is so small that it may be merely a result of errors of the tide-gauge. The solar semiannual (seven-hundredths of a foot above and below mean level) giving highest average level Feb. 14 and Aug. 15, seems too large to be not genuine; but it cannot be astronomical, or there would be a corresponding lunar tide.

46. The solar annual (referred to in § 10), as shown by the calculations, (Ramsgate, year 1864, being 13 of a foot above and below mean), is certainly much too large to be attributable to the eccentricity of the earth's orbit, and the time of its maximum (Sept. 21) does not at all suit the astronomical theory. Its origin (as well as that of the semiannual?) is in all probability meteorological. The Liverpool observations for 1857-58 show a greater difference (36 of a foot above and below mean level); and time of maximum average height, Oct. 20.

Progress after date of Mr. Roberts's Supplementary Report.

47. The deduction of the lunar and solar semidiurnal and diurnal tides from the Fiji observations (§§ 26, 38), which is no doubt practicable, is a mathematical problem of considerable interest. A good deal of work towards it has been performed by Mr. Roberts since the date of the conclusion of his Supplementary Report. The plan followed has been simply a direct application of the method of least squares, as in § 28, and it has been carried out so far as the formation of eleven simple equations for the determination of eleven unknown quantities, viz. :—